Continuous Stochastic Calculus with Applications to Finance APPLIED MATHEMATICS Editor: R.J Knops This series presents texts and monographs at graduate and research level covering a wide variety of topics of current research interest in modern and traditional applied mathematics, in numerical analysis and computation Introduction to the Thermodynamics of Solids J.L Ericksen (1991) Order Stars A Iserles and S.P Nørsett (1991) Material Inhomogeneities in Elasticity G Maugin (1993) Bivectors and Waves in Mechanics and Optics Ph Boulanger and M Hayes (1993) Mathematical Modelling of Inelastic Deformation J.F Besseling and E van der Geissen (1993) Vortex Structures in a Stratified Fluid: Order from Chaos Sergey I Voropayev and Yakov D Afanasyev (1994) Numerical Hamiltonian Problems J.M Sanz-Serna and M.P Calvo (1994) Variational Theories for Liquid Crystals E.G Virga (1994) Asymptotic Treatment of Differential Equations A Georgescu (1995) 10 Plasma Physics Theory A Sitenko and V Malnev (1995) 11 Wavelets and Multiscale Signal Processing A Cohen and R.D Ryan (1995) 12 Numerical Solution of Convection-Diffusion Problems K.W Morton (1996) 13 Weak and Measure-valued Solutions to Evolutionary PDEs J Málek, J Necas, M Rokyta and M Ruzicka (1996) 14 Nonlinear Ill-Posed Problems A.N Tikhonov, A.S Leonov and A.G Yagola (1998) 15 Mathematical Models in Boundary Layer Theory O.A Oleinik and V.M Samokhin (1999) 16 Robust Computational Techniques for Boundary Layers P.A Farrell, A.F Hegarty, J.J.H Miller, E O’Riordan and G I Shishkin (2000) 17 Continuous Stochastic Calculus with Applications to Finance M Meyer (2001) (Full details concerning this series, and more information on titles in preparation are available from the publisher.) Continuous Stochastic Calculus with Applications to Finance MICHAEL MEYER, Ph.D CHAPMAN & HALL/CRC Boca Raton London New York Washington, D.C Library of Congress Cataloging-in-Publication Data Meyer, Michael (Michael J.) Continuous stochastic calculus with applications to finance / Michael Meyer p cm. (Applied mathematics ; 17) Includes bibliographical references and index ISBN 1-58488-234-4 (alk paper) Finance Mathematical models Stochastic analysis I Title II Series HG173 M49 2000 332′.01′5118—dc21 00-064361 This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale Specific permission must be obtained in writing from CRC Press LLC for such copying Direct all inquiries to CRC Press LLC, 2000 N.W Corporate Blvd., Boca Raton, Florida 33431 Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe © 2001 by Chapman & Hall/CRC No claim to original U.S Government works International Standard Book Number 1-58488-234-4 Library of Congress Card Number 00-064361 Printed in the United States of America Printed on acid-free paper Preface v PREFACE The current, prolonged boom in the US and European stock markets has increased interest in the mathematics of security markets most notably the theory of stochastic integration Existing books on the subject seem to belong to one of two classes On the one hand there are rigorous accounts which develop the theory to great depth without particular interest in finance and which make great demands on the prerequisite knowledge and mathematical maturity of the reader On the other hand treatments which are aimed at application to finance are often of a nontechnical nature providing the reader with little more than an ability to manipulate symbols to which no meaning can be attached The present book gives a rigorous development of the theory of stochastic integration as it applies to the valuation of derivative securities It is hoped that a satisfactory balance between aesthetic appeal, degree of generality, depth and ease of reading is achieved Prerequisites are minimal For the most part a basic knowledge of measure theoretic probability and Hilbert space theory is sufficient Slightly more advanced functional analysis (Banach Alaoglu theorem) is used only once The development begins with the theory of discrete time martingales, in itself a charming subject From these humble origins we develop all the necessary tools to construct the stochastic integral with respect to a general continuous semimartingale The limitation to continuous integrators greatly simplifies the exposition while still providing a reasonable degree of generality A leisurely pace is assumed throughout, proofs are presented in complete detail and a certain amount of redundancy is maintained in the writing, all with a view to make the reading as effortless and enjoyable as possible The book is split into four chapters numbered I, II, III, IV Each chapter has sections 1,2,3 etc and each section subsections a,b,c etc Items within subsections are numbered 1,2,3 etc again Thus III.4.a.2 refers to item in subsection a of section of Chapter III However from within Chapter III this item would be referred to as 4.a.2 Displayed equations are numbered (0), (1), (2) etc Thus II.3.b.eq.(5) refers to equation (5) of subsection b of section of Chapter II This same equation would be referred to as 3.b.eq.(5) from within Chapter II and as (5) from within the subsection wherein it occurs Very little is new or original and much of the material is standard and can be found in many books The following sources have been used: [Ca,Cb] I.5.b.1, I.5.b.2, I.7.b.0, I.7.b.1; [CRS] I.2.b, I.4.a.2, I.4.b.0; [CW] III.2.e.0, III.3.e.1, III.2.e.3; vi Preface [DD] II.1.a.6, II.2.a.1, II.2.a.2; [DF] IV.3.e; [DT] I.8.a.6, II.2.e.7, II.2.e.9, III.4.b.3, III.5.b.2; [J] III.3.c.4, IV.3.c.3, IV.3.c.4, IV.3.d, IV.5.e, IV.5.h; [K] II.1.a, II.1.b; [KS] I.9.d, III.4.c.5, III.4.d.0, III.5.a.3, III.5.c.4, III.5.f.1, IV.1.c.3; [MR] IV.4.d.0, IV.5.g, IV.5.j; [RY] I.9.b, I.9.c, III.2.a.2, III.2.d.5 vii To my mother ix Table of Contents TABLE OF CONTENTS Chapter I Martingale Theory Preliminaries 1 Convergence of Random Variables 1.a Forms of convergence 1.b Norm convergence and uniform integrability Conditioning 2.a Sigma fields, information and conditional expectation 2.b Conditional expectation 10 Submartingales 19 3.a Adapted stochastic processes 19 3.b Sampling at optional times 22 3.c Application to the gambler’s ruin problem 25 Convergence Theorems 29 4.a Upcrossings 4.b Reversed submartingales 4.c Levi’s Theorem 4.d Strong Law of Large Numbers 29 34 36 38 Optional Sampling of Closed Submartingale Sequences 42 5.a Uniform integrability, last elements, closure 42 5.b Sampling of closed submartingale sequences 44 Maximal Inequalities for Submartingale Sequences 47 6.a Expectations as Lebesgue integrals 47 6.b Maximal inequalities for submartingale sequences 47 Continuous Time Martingales 50 7.a Filtration, optional times, sampling 7.b Pathwise continuity 7.c Convergence theorems 7.d Optional sampling theorem 7.e Continuous time Lp -inequalities Local Martingales 50 56 59 62 64 65 8.a Localization 65 8.b Bayes Theorem 71 306 D Kolmogoroff Existence Theorem By contrast to the case of complex scalars this does not imply that the matrix C is symmetric, as the example of the matrix C = 10 21 shows If C is symmetric and positive semidefinite, then λj = λj fj = (λj fj , fj ) = (Cfj , fj ) ≥ and it follows that λ1 , λ2 , λk > and λk+1 = λk+2 = = λn = It is now easily seen that C can be written as C = QQ , for some n × n matrix Q Indeed Q = U diag( λj ) yields such a matrix Q: From U −1 CU = diag(λj ) it follows that C = U diag(λj )U −1 = U diag(λj )U = U diag( λj ) U diag( λj ) = QQ Indeed it is even true that C has a positive squareroot (i.e., Q above can be chosen to be symmetric and positive semidefinite) We not need this The relation C = QQ will be the key in the proof of the existence of Gaussian random variables with arbitrary parameters m ∈ Rn and C a symmetric, positive semidefinite n × n matrix Let us note that the matrix Q = U diag( λj ) satisfies range(Q) = range(C) Indeed, using the equality CU = U diag(λj ), we have range(C) = range(CU ) = range(U diag(λk )) = span{ U e1 , U e2 , , U ek } = range(Q) D Kolmogoroff Existence Theorem Compact classes and countable additivity Let E be a set A family K0 of subsets of E has the f inite intersection property, if K0 ∩ K1 ∩ ∩ Kn = ∅, for each finite subfamily {K0 , K1 , , Kn } ⊆ K0 A compact class on E is now a family K of subsets of E such that K∈K0 K = ∅, for each subfamily K0 ⊆ K, which has the finite intersection property, that is, K0 ⊆ K and K0 ∩ ∩ Kn = ∅, for each finite subfamily {K0 , , Kn } ⊆ K0 , ⇒ K∈K0 K = ∅ In more familiar terms: If we set S = { K = E \ K : K ∈ K }, then K is a compact class on E if and only if every cover of E by sets in S has a finite subcover Thus the family of closed subsets of a compact topological space E is always a compact class on E Similarly the family of all compact subsets of a Hausdorff space E is also a compact class on E Here the Hausdorff property is needed to make all compact sets closed These compact classes are closed under finite unions c D.1 Let K be a compact class on the set E Then (a) Every subfamily of K is again a compact class on E (b) There is a topology on E in which E is compact and such that K is contained in the family of all closed subsets of E (c) There is a compact class K1 on E such that K ⊆ K1 and K1 is closed under finite unions (d) The family of all finite unions of sets in K is again a compact class on E Proof (a) is clear (b) Set S = { K c | K ∈ K }, then every cover of E by sets in K has a finite subcover By the Alexander Subbasis Theorem E is compact in the topology generated by S as a subbasis on E Clearly every set K ∈ K is closed in this topology (c) Let K1 be the family of all closed subsets of E in the topology of (c) (d) follows from (c) and (a) Appendix 307 D.2 Let A be a field of subsets of E, µ : A → [0, +∞) a finite, finitely additive set function on A and K ⊆ A a compact class on E If µ is inner regular with respect to K in the sense that µ(A) = sup{ µ(K) | K ∈ K, K ⊆ A }, for all sets A ∈ A, then µ is countably additive on A Proof Since µ is already finitely additive and µ(E) < +∞, the countable additivity of µ is implied by the following condition: (Dn )∞ n=1 ⊆ A, Dn ↓ ∅ ⇒ µ(Dn ) ↓ (0) To verify (0) consider such a sequence (Dn )∞ n=1 ⊆ A and let > be arbitrary For each n ≥ choose a set Kn ∈ K such that Kn ⊆ Dn As that and µ(Dn \ Kn ) < /2n Dn = ∅ it follows that n≥1 Kn = ∅ Since K is a compact class, it follows n≤N Kn = ∅, for some finite number N Then n ≥ N implies n≥1 Dn ⊆ DN = DN \ and so µ(Dn ) ≤ N j=1 N j=1 Kj = µ(Dj \ Kj ) < N j=1 (DN N j=1 \ Kj ) ⊆ N j=1 (Dj \ Kj ) /2j < Thus µ(Dn ) → 0, as n ↑ ∞ Products Let E be a compact Hausdorff space, E the Borel σ-field on E and I any index set The product Ω = E I is then the family of all functions ω : I → E We write ω(t) = ωt , t ∈ I, and ω = (ωt )t∈I Equipped with the product topology Ω is again a compact Hausdorff space by Tychonoff’s Theorem For t ∈ I we have the projection (coordinate map) πt : ω ∈ Ω → ω(t) ∈ E The product σ-field E I on Ω = E I is then defined to be the σ-field σ(πt , t ∈ I) generated by the coordinate maps πt It is characterized by the following universal property: a map X from any measurable space into E I , E I is measurable if and only if πt ◦ X is measurable for each t ∈ I More generally, for all subsets H ⊆ J ⊆ I, we have the natural projections πH : Ω = E I → ΩH = E H and πJH : ΩJ = E J → ΩH = E H which are measurable with respect to the product σ-fields and satisfy πH = πJH ◦ πJ , H ⊆ J ⊆ I In this notation πt = πH , where H = {t} Let H(I) denote the family of all f inite subsets of I For each set H = {t1 , t2 , , tn } ∈ H(I) we have πH (ω) = (ωt1 , ωt2 , , ωtn ) ∈ ΩH = E H 308 D Kolmogoroff Existence Theorem −1 If H ∈ H(I) and BH ∈ EH , then the subset Z = πH (BH ) ⊆ Ω is called the f inite dimensional cylinder with base BH This cylinder is said to be represented on the −1 set H ∈ H(I) The cylinder Z = πH (BH ) also satisfies −1 −1 Z = πH (BH ) = πJ−1 πJH (BH ) = πJ−1 (BJ ), −1 −1 (BH ) ∈ EJ In other words, the cylinder Z = πH (BH ) can be where BJ = πJH represented on every set J ∈ H(I) with J ⊇ H Thus any two cylinders Z1 , Z2 can be represented on the same set H ∈ H(I) Since −1 (BH ) πH c −1 −1 −1 −1 c = πH (BH ) and πH (BH ) ∩ πH (CH ) = πH (BH ∩ CH ) −1 it follows that the family Z = { πH (BH ) ⊆ Ω | H ∈ H(I) and BH ∈ EH } of finite dimensional cylinders is a field of subsets of Ω Clearly the finite dimensional cylinders generate the product σ-field E I If we merely need a π-system of generators for the product σ-field E I we can manage with a far smaller family of sets A f inite dimensional rectangle is a set Z of the form Z = t∈H πt−1 (Et ) = t∈H πt ∈ Et , where H ∈ H(I) and Et ∈ E, for all t ∈ H Thus Z is the cylinder based on the rectangle BH = t∈H Et ∈ E H The finite dimensional rectangles in E I no longer form a field but they are still a π-system generating the product σ-field E I Indeed, the set H in the definition of Z can be enlarged, by setting Et = E for the new elements t, without altering Z Thus any two finite dimensional rectangles can be represented on the same set H ∈ H(I) and from this it follows easily that the intersection of any two finite dimensional rectangles is another such rectangle Each finite dimensional rectangle is in E I and thus the σ-field G generated by the finite dimensional rectangles satisfies G ⊆ E I On the other hand each coordinate map πt is G measurable and this implies E I ⊆ G Thus E I = G Finite dimensional rectangles are extremely basic events and a σ-field on the product space E I will not be useful unless it contains them all In this sense the product σ-field E I is the smallest useful σ-field on E I It has the following desirable property: a probability measure Q on E I is uniquely determined by its values on finite dimensional rectangles in E I Usually, when such a measure Q is to be constructed to reflect some probabilistic intuition, it is clear what Q has to be on finite dimensional rectangles and this then determines Q on all of E I If such uniqueness does not hold, the problem arises which among all possible candidates best reflects the underlying probabilistic intuition The product topology on Ω = E I provides us with two more σ-fields on Ω, the Baire σ-field (the σ-field generated by the continuous (real valued) functions on Ω) and the Borel σ-field B(Ω) (the σ-field generated by the open subsets of Ω) Let us say that a function f = f (ω) on Ω depends only on countably many coordinates of the point ω = (ωt )t∈I ∈ Ω if there exists a countable subset I0 ⊆ I such that f (ω) = f (˜ ω ), for all ω, ω ˜ ∈ Ω with ω|I0 = ω ˜ |I0 Likewise a subset A ⊆ Ω is said to depend only on countably many coordinates if this is true of its indicator function 1A , equivalently, if there exists a countable subset I0 ⊆ I such that ω ∈ A ⇐⇒ ω ˜ ∈ A, for all ω, ω ˜ ∈ Ω with ω|I0 = ω ˜ |I0 Appendix 309 D.3 (a) The product σ-field E I is the Baire σ-field on Ω and thus is contained in the Borel σ-field B(Ω) If I is countable, then B(Ω) = E I (b) Every set A in the product σ-field E I depends only on countably many coordinates Thus, if I is uncountable, then B(Ω) = E I (c) Every function f = f (ω) : Ω → R which is measurable with respect to the product σ-field E I on Ω depends only on countably many coordinates of the point ω = (ωt )t∈I ∈ Ω In particular this is true for all continuous functions on Ω Proof (a) Since each coordinate map πt is continuous, the product σ-field E I is contained in the Baire σ-field on Ω To see the reverse inclusion let C r (Ω) denote the real algebra of all continuous functions f : Ω → R It will suffice to show that every function f ∈ C r (Ω) is in fact measurable with respect to the product σ-field E I This is certainly true of the projections πt , t ∈ I, and of the constant function and hence of every function in the subalgebra A ⊆ C r (Ω) generated by these functions By the measurability of pointwise limits, every pointwise limit of functions in A is measurable for the product σ-field E I on Ω Thus it remains to be shown only that every function f ∈ C r (Ω) can be represented as a pointwise limit of functions in A In fact the subalgebra A ⊆ C r (Ω) separates points on Ω and contains the constants and is thus even uniformly dense in C r (Ω), by the Stone Weierstrass Theorem Assume now that I is countable Then so is the family H(I) The family −1 G = { πH (GH ) | H ∈ H(I), GH ⊆ ΩH open } (1) is a basis for the product topology on Ω An arbitrary union of such basic sets with fixed index set H is a set of the same form Thus any open set G ⊆ Ω can be written as a union of basic open sets with distinct index sets H and any such union is necessarily countable, by the countability of H(I) Thus, if G is any open subset of Ω, then G is a countable union of basic open sets as in (1) Each such basic open set is a finite dimensional cylinder and hence in the product σ-field E I Thus G ∈ E I and it follows that B(Ω) ⊆ E I (b) The family F0 of all subsets A ⊆ Ω, which depend only on countably many coordinates, is easily seen to be a σ-field containing all finite dimensional cylin−1 ders Z = πH (BH ), H ∈ H(I), BH ∈ EH (such a cylinder depends only on the coordinates in the set H) and hence the entire product σ-field E I Note now that a singleton set A = {ω} ⊆ Ω depends on all coordinates Thus, if I is uncountable, then the product σ-field E I contains no singleton sets The Borel σ-field B(Ω) on the other hand contains all singleton sets since these are closed (the product Ω = E I is again Hausdorff) (c) The family C of all nonnegative functions f : Ω → R which depend on only finitely many coordinates is easily seen to be a λ-cone on Ω containing the indicator function of every finite dimensional rectangle (see (b)) Since these rectangles form 310 D Kolmogoroff Existence Theorem a π-system generating the product σ-field E I , C contains every nonnegative, E I measurable function f : Ω → R (B.4) The extension to arbitrary E I -measurable f : Ω → R is trivial Projective families of probability measures Let P be a probability measure on the product space (Ω, E) = (E I , E I ) For each subset H ∈ H(I) the image measure PH = πH (P ) is defined on the finite product (ΩH , EH ) = (E H , E H ) as −1 PH (BH ) = P (πH (BH )), for all sets BH ∈ EH The measures PH , H ∈ H(I), are called the f inite dimensional marginal distributions of P The relation πH = πJH ◦ πJ implies the following consistency relation for these marginal distributions: πJH (PJ ) = πJH (πJ (P )) = πH (P ) = PH , for all H, J ∈ H(I) with H ⊂ J (2) Conversely assume that for each set H ∈ H(I), PH is a probability measure on the the finite product (E H , E H ) We wonder whether there exists a probability measure P on the product space (E I , E I ) such that πH (P ) = PH , for all H ∈ H(I), that is such that the finite dimensional marginal distributions of P are the measures PH , H ∈ H(I) Clearly such a measure P can exist only if the measures PH satisfy the consistency relation (2) If this relation is satisfied we call the family (PH )H∈H(I) projective The projective property of the measures (PH )H∈H(I) is also sufficient for the existence of the measure P on E I under very general assumptions We not need this result in full generality The proof can be simplified greatly if suitable conditions are imposed on the measurable space (E, E) D.4 DeÞnition The measurable space (E, E) is called standard if there exists a metric on E which makes E compact and E the Borel σ-field on E Remark If there exists a bimeasurable isomorphism φ : (T, B(T )) → (E, E), where T is any compact metric space and B(T ) the Borel σ-field on T , then obviously (E, E) is a standard measurable space We will now exhibit such an isomorphism φ : (T, B(T )) → (R, B(R)), where T = [−1, 1] Such φ induces an isomorphism φd : T d , B(T d ) → Rd , B(Rd ) We will then have shown that Rd , B(Rd ) is a standard measurable space, for all d ≥ Set An = [−1/n, −1/(n + 1)[ ∪ ]1/(n + 1), 1/n], Bn = [−n, −(n − 1)[ ∪ ]n − 1, n] and φn (t) = n(n + 1)t − sgn(t), t ∈ An , n ≥ 1, where sgn(t) = +1, if t > 0, and sgn(t) = −1, if t < 0, as usual Note that φn : An → Bn is a bimeasurable isomorphism, for all n ≥ Moreover T = [−1, 1] is the disjoint union {0} ∪ An and R is the disjoint union {0} ∪ Bn Set φ = 1An φn , that is, φ : T → R satisfies φ(0) = and φ = φn on An It follows that φ : (T, B(T )) → (R, B(R)) is a bimeasurable isomorphism Appendix 311 D.5 Let T be a compact metric space and P a probability measure on the Borel σ-field B(T ) of T Then every Borel set A ⊆ T satisfies P (A) = sup{ P (K) | Kcompact, K ⊆ A } (3) Proof It is not hard to show that the family G of all Borel sets B ⊆ T such that both sets A = B, B c satisfy (3) is a σ-field [DD, 7.1.2] Any open set G ⊆ T is the increasing union of the compact sets Kn = { x ∈ T | dist(x, Gc ) ≥ 1/n } and consequently P (Kn ) ↑ P (G), n ↑ ∞ Thus G ∈ G It follows that G contains every Borel subset of T D.6 Kolmogoroff Existence Theorem Let (E, E) be a standard measurable space If (PH )H∈H(I) is a projective family of probability measures on (ΩH , EH ) H∈H(I) then there exists a unique probability measure P on the product space (E I , E I ) such that PH = πH (P ), for all sets H ∈ H(I) Proof The condition PH = πH (P ), for all sets H ∈ H(I), uniquely determines P on cylinders and hence on the entire product σ-field E I (the cylinders form a π-system of generators for E I ) To see the existence of P define the set function P0 on the field Z of finite dimensional cylinders as P0 (Z) = PH (BH ), −1 Z = πH (BH ) ∈ Z, H ∈ H(I), BH ∈ EH −1 −1 (BH ) = πK (BK ) with It must be shown that P0 is well defined Indeed if Z = πH sets H, K ∈ H(I) and BH ∈ EH , BK ∈ EK , set J = H ∪ K ∈ H(I) and note that −1 −1 −1 −1 Z = πH (BH ) = πJ−1 πJH (BH ) and likewise Z = πK (BK ) = πJ−1 πJK (BK ) and thus −1 −1 πJ−1 πJH (BH ) = πJ−1 πJK (BK ) −1 −1 Since the projection πJ is surjective, it follows that πJH (BH ) = πJK (BK ) and consequently, by the projective property of the measures PH , H ∈ H(I), −1 −1 PH (BH ) = PJ πJH (BH ) = PJ πJK (BK ) = PK (BK ), as desired Clearly P0 (Ω) = It will now suffice to show that the set function P0 : Z → [0, 1] is countably additive, for then it extends to a probability measure P on the σ-field E I generated by the field Z of cylinders and this extension obviously satisfies PH = πH (P ), for all sets H ∈ H(I) Clearly P0 is finitely additive on Z To see countable additivity, equip E with a metric with which it becomes a compact space such that E is the Borel σ-field on E Then each finite product ΩH = E H is compact and EH = E H = B(ΩH ) is the Borel σ-field on ΩH Moreover the Borel probability measure PH on ΩH is automatically inner regular with respect to the family of compact subsets of ΩH −1 (D.5) Thus for each cylinder Z = πH (BH ) ∈ Z, H ∈ H(I), BH ∈ EH , we have P0 (Z) = PH (BH ) = sup{ PH (KH ) | KH ⊆ BH , KH compact } −1 (KH )) | KH ⊆ BH , KH compact } = sup{ P0 (πH 312 D Kolmogoroff Existence Theorem This shows that P0 is inner regular with respect to the family −1 K = { πH (KH ) | H ∈ H(I), KH ⊆ BH , KH compact } ⊆ Z −1 If H ∈ H(I) and KH is a compact subset of BH then the set πH (KH ) is closed in I the compact product space Ω = E and hence itself compact Thus K is contained in the family of compact subsets of Ω and hence is a compact class The countable additivity of P0 now follows according to D.1 Bibliography 313 BIBLIOGRAPHY [Ca] [Cb] K L CHUNG (1974), A Course in Probability Theory, Academic Press K L CHUNG (1974), Lectures from Markov Processes to Brownian Motion, Heidelberg, Springer-Verlag [CRS] Y CHOW, H ROBBINS, D SIEGMUND (1991), The Theory of Optimal Stopping, New York, Dover [CW] K.L CHUNG, R J WILLIAMS (1990), Introduction to Stochastic Integration, New York, Springer-Verlag [DB] F DELBAEN, Representing Martingale Measures when Asset Prices are Continuous and Bounded,Mathematical Finance Vol 2, No (1992), 107-130 [DD] R M DUDLEY (1989), Real Analysis and Probability, Belmont, WadsworthBrooks/Cole [DF] D DUFFIE (1996), Dynamic Asset Pricing Theory, New Haven, Princeton University Press [DS] F DELBAEN, W SCHACHERMAYER, A General Version of the Fundamental Theorem of Asset Pricing, Math Ann 300, 463-520 [DT] R DURRETT (1996), Stochastic Calculus, a Practical Introduction, Boca Raton, CRC Press [J] F JAMSHIDIAN, Libor and Swap Market Models and Measures, Finance and Stochastics (1997), 293-330 [K] N.V KRYLOV (1995), Introduction to the Theory of Diffusion Processes, Providence, American Mathematical Society [KS] I KARATZAS, S SHREVE (1988), Brownian Motion and Stochastic Calculus, Heidelberg, Springer-Verlag [MR] M MUSIELA, M RUTKOVSKI (1998), Martingale Methods in Financial Modelling, Heidelberg, Springer-Verlag [P] P PROTTER (1990), Stochastic Integration and Differential Equations, Heidelberg, Springer-Verlag [PTH] K R PARTHASARATHY (1967), Probability Measures on Metric Spaces, New York, Academic Press [Ra] W RUDIN (1966), Real and Complex Analysis, New York, McGraw-Hill [Rb] W RUDIN (1973), Functional Analysis, New York, McGraw-Hill [RY] D REVUZ, M YOR (1999), Continuous Martingales and Brownian Motion, Heidelberg, Springer-Verlag Index INDEX Adapted process 19 Almost surely Approximation of sets by generators 304 Arbitrage price 253 Arbitrage strategy 215, 231 p-approximate 241 Assumption for filtrations 50 Gaussian 259 Baire σ -field 309 Barrier option 272 Bayes Theorem 71 Black Scholes Formula 223, 226 Market 211 Bond fixed rate 276 floating rate 276 zero coupon 243 Borel Cantelli Lemma Borel σ -field 1, 309 Bounded variation asset, see riskless bond portfolio 215 process 91, 98 Brownian functional 187 local martingale 192 Brownian motion 112, 118 augmented filtration 127 martingale characterization 166 quadratic variation 85, 166 Call 221 Cap, interest rate 280 Caplet 280 Change of measure 170 of numeraire 236 claim (option) 218, 252 replication 218, 257, 265, 269, 271 valuation, see pricing Clark’s separation theorem 298 315 Class DL 68 Closed submartingale 42 Compact class 306 Compensator additive 98 multiplicative 168 Conditional expectation and independence 11, 17 Computation 15 Definition 10 Dominated Convergence 15, 37 Fatou’s lemma 14 Heuristics Monotone Convergence 14 Consistency of finite dimensional distributions 309 Continuity of Brownian local martingales 194 Continuous process 50 version 58, 194 Convergence modes of Theorems 14, 15, 31-34, 37, 59, 61 Convex cone 296 function of a martingale 20 sets, separation 298 Countable dependence on coordinates 309 product 308 σ -field 12 Covariance matrix 104 Covariation process 90, 99 Cylinder sets 308 Decomposition Kunita-Watanabe 196 semimartingale 98 Deflator 232 Delta hedging 265 Density process 21, 71, 170 standard normal 103 Derivative security 218 Differentials, stochastic 160 Discretization Lemma 54 Distribution Gaussian 106 316 Index of a random object 1, 301 standard normal 103 Dividends 225, 271 Doleans exponential 173 measure µM 135 Dominated Convergence Theorem for conditional expectation 15, 37 for stochastic integrals 149 Drift process 200 Dynamic portfolio, see trading strategy Dynamics Ito process 200 Libor process 281 Swap rate process 286 Equality of processes 50 Equivalent martingale measure 212, 227, 235, 238 European option 252 Call, Put 223, 226 of form f (ST ) 221 valuation PDE 267-271 Evaluation functional 116 Exercise set 254 Expected value as a Lebesgue integral 47 Exponential equation 173 Exponential local martingale 173 Extension basic procedure 299 of positive functionals 240 Theorem 301 Fatou’s Lemma 14 Filtration 19 augmented 19 generated by Brownian motion 127 right continuous 50 standard assumptions 50 Finite dimensional cylinders 308 distributions 309 rectangle 308 Forward cap 280 contract 221, 254 Libor rate 245 martingale measure 245 price 220, 246, 250 swap rate 277 swaption 279 Free lunch with vanishing risk 242 Fubini’s theorem 303 Functional positive 239, 298 square integrable 187 Gambler’s ruin 25 Gaussian distribution, random variables 106 process 109 Girsanov’s formula 170 theorem 175, 178 Harmonic function 169 Hedging strategy, see replicating strategy Hitting time 55 Image measure 301 Increments independent 112, 118 stationary 113 Independence 303 Indistinguishable processes 50 Inequality Jensen 17 Kunita-Watanabe 97 Lp 48, 64 Maximal 47, 64 Upcrossing 31 Integrability, uniform Integral representation 187-196 stochastic 91, 137, 141, 147 with respect to a Gaussian law 109 Interest rates 243-245 forward 244 Libor 245 Swap rate 277 Interest rate swap 277 Index Isonormal process 111 Ito formula 157, 159 process 200 Jensen’s inequality 17 Kunita-Watanabe decomposition 196 inequality 97 Kolmogoroff Existence Theorem 306 Zero One Law 39 λ-cone 303 λ-system 301 Last element 42 Law of One Price 217, 252 L2 -bounded martingale 73, 86 Lp -bounded family 4, Lp -inequality 48, 64 Left continuous process 50, 133 Levi Characterization of Brownian Convergence Theorem 36 motion 166 Libor model 282 rate 245 process dynamics 281 Local martingale 66 Localization 65 Locally equivalent probability 71 Logarithmic transformation 200 Log-Gaussian process 208 Market 228 Market price of risk 247 Markov property 120, 125 Martingale 19 closed 42, 61, 62 convergence theorem 33, 61 convex function of 20 Gaussian 208 last element 42 L2 -bounded 74, 87 317 representation theorems 183-196 right continuous version 58 square integrable 73, 86 uniformly integrable 42, 62 Measurability properties of processes 131 with respect to σ(X) 302 Monotone Convergence Theorem 14 Multinormal distribution, see Gaussian distribution Normal distribution, see Gaussian distribution Novikov condition 175, 180, 182 Null sets 17, 19, 127 Numeraire asset 235 change 236 invariance 231 measure 235 Option 218, 252 digital 263 martingale pricing 218, 253 power 264 to exchange assets 254, 262 Optional process 134 Sampling Theorem 24, 45, 62 σ -field 134 time 22, 51 Orthogonal martingales 196 Partial differential equations 267-271 Path 50 Pathspace 116, 119 Pathwise properties 56 Payer swap 277 π -λ-Theorem 301 π -system 299 Positive Brownian martingale 196 functional 239 semidefinite matrix 305 Predictable process 131 σ -field 131 318 Index Price functional 239 Pricing assumption 260 cap 285 European puts and calls 223, 225 European options under Gaussian floor 285 forward contract 254 martingale 218, 253 option to exchange assets 254, 262 PDE 267-271 swap 277 swaption 291 Product measure 310 σ -field 308 space 307 Progressive measurability of a process 92 σ -field 131 Projectve family of measures 310 Put 221 Put Call Parity 221, 227 Quadratic variation 73 and L2 -bounded martingales 86 and L1 -bounded martingales 88 of a Brownian motion 85, 166 of a local martingale 82 of a semimartingale 99 Random time 22 Random variable, vector, object Rates, see interest rates Receiver swap 275 Reducing a process 66 Reducing sequence 66 Replication 218, 257, 265, 269, 271 Replicating strategy 218, 252 Representation of martingales as stochastic integrals 192-196 as time change of Brownian motion 185 Reset date 276 Reversed submartingale 34 Right continuous filtration 50 version 58 Riskfree rate 211 Riskless bond 237 portfolio 215 Risk neutral valuation 218, 253 Sampling of a process 22, 53 Scaling 129 Self-financing property 214, 230 numeraire invariance 231 Semimartingale 98 vector valued 153 Separation of convex sets 297 Settlement date 276 Shift operators 119 Short rate process 247 σ -field Baire 309 Borel 1, 309 FT 22, 51 optional 134 predictable 131 product 306 progressive 131 Simple predictable process 132 Square integrable functional 187 martingale 73, 86 Spot martingale measure 237 Stochastic differential 160 integral 91, 137, 141, 147 interest rates 243-245, 247 interval 95 process 19 product rule 151 Stock price 165 Stopping a process 65 Stopping time, see optional time Strike price 221 Strong Law of Large Numbers 38 Strongly arbitrage free market 241 Submartingale 19 closeable, last element 42, 62 Convergence theorem 31, 59 Supermartingale 19 Swap (interest rate) 277 Swap rate model 288 Swaption 280 Index Symmetric numeraire change formula 236 Tenor 276 Term structure of interest rates 243-245 Time change for local martingales 183 hitting 55 optional 22, 51 random 22 Trading strategy 213, 229 arbitrage 215, 231 nonnegative 231 replicating 218, 252 self-financing 213, 225, 230 tame 213, 230 Uniformly integrable family of random variables martingale 42 Upcrossing 29 Lemma 31 Usual conditions for a filtration 50 Valuation, see pricing Version of a process 50 right continuous 58 Volatility 200, 203 Wald’s identity 179 Wiener measure 117 process, see Brownian motion Zero coupon bond 243 Zero One Law, Kolmogoroff 39 319 [...]... Properties of stochastic integrals with respect to continuous local martingales 2.d Integration with respect to continuous semimartingales 2.e The stochastic integral as a limit of certain Riemann type sums 2.f Integration with respect to vector valued continuous semimartingales 135 140 142 147 150 153 Table of Contents 3 Ito’s Formula xi 157 3.a Ito’s... 112 116 118 118 120 127 128 Chapter III Stochastic Integration 1 Measurability Properties of Stochastic Processes 131 1.a The progressive and predictable σ-fields on Π 131 1.b Stochastic intervals and the optional σ-field 134 2 Stochastic Integration with Respect to Continuous Semimartingales 135 2.a Integration with respect to continuous local martingales 2.b M -integrable... expectation of X with respect to G) Processes Let X = (Xt )t≥0 be a stochastic process and T : Ω → [0, ∞] an optional time Then XT denotes the random variable (XT )(ω) = XT (ω) (ω) (sample of X along T , I.3.b, I.7.a) X T denotes the process XtT = Xt∧T (process X stopped at time T ) S, S+ and S n denote the space of continuous semimartingales, continuous positive semimartingales and continuous Rn -valued... the set D in the proof of (g) is necessary since the σ-field G is not assumed to contain the null sets Since E(P) is not a vector space, EG : X ∈ E(P ) → EG (X) is not a linear operator However when its domain is restricted to L1 (P ), then EG becomes a nonnegative linear operator 14 2.b Conditional expectation 2.b.6 Monotone Convergence Let Xn , X, h ∈ E(P ) and assume that Xn ≥ h, n ≥ 1, and Xn ↑... ∈ I, is uniformly absolutely continuous with respect to the measure P (c) From 1.b.0 it follows that each finite family F = { f1 , f2 , , fn } ⊆ L1 (P ) of integrable functions is both uniformly integrable (increase c) and uniformly P continuous (decrease δ) 1.b.2 A family F = { Xi | i ∈ I } of random variables is uniformly integrable if and only if F is uniformly P -continuous and L1 -bounded Proof... function of c ≥ 0 Consequently, to show that the family F = { Xi | i ∈ I } is uniformly integrable it suffices to show that for each > 0 there exists a c ≥ 0 such that supi∈I E |Xi |; [|Xi | ≥ c] ≤ (b) To show that the family F = { Xi | i ∈ I } is uniformly P -continuous we must show that for each > 0 there exists a δ > 0 such that supi∈I E 1A |Xi | < , for all sets A ∈ F with P (A) < δ This means that... for P -ae ω ∈ Ω t 0 Hs dAs is defined pathwise Assume now that X is a continuous semimartingale with semimartingale decomposition X = A + M (A = uX , M a continuous local martingale, I.11.a) Then L(X) = L1loc (A) ∩ L2loc (M ) Thus L(X) = L2loc (X), if X is a local martingale For H ∈ L(X) set H • X = H • A+H • M Then H • X is the unique continuous semimartingale satisfying (H • X)0 = 0, uH • X = H • uX... the integrand H is continuous we have the t 0 Hs dXs = lim ∆ →0 S∆ (H, X) (limit in probability), where S∆ (H, X) = Htj−1 (Xtj − Xtj−1 ) for ∆ as above (III.2.e.0) The (deterministic) process t defined by t(t) = t, t ≥ 0, is a continuous semimartingale, in fact a bounded variation process Thus the spaces L(t) and L1loc (t) are defined and in fact L(t) = L1loc (t) Vector valued integrators Let X ∈ S d and... (X 1 , X 2 , , X d ) (column vector), with X j ∈ S Then L(X) is the space of all Rd -valued processes H = (H 1 , H 2 , , H d ) such that H j ∈ L(X j ), for all j = 1, 2, , d For H ∈ L(X), H •X = j Hj •Xj, (H • X)t = dX = (dX 1 , dX 2 , , dX d ) , t 0 Hs · dXs = Hs · dXs = j j t 0 Hsj dXsj , Hsj dXsj If X is a continuous local martingale (all the X j continuous local martingales), the spaces... L(X) and K • Z = H • X (III.3.b) With the process t as above we have dt(t) = dt Local martingale exponential Let M be a continuous, real valued local martingale Then the local martingale exponential E(M ) is the process Xt = Et (M ) = exp Mt − 1 2 M t X = E(M ) is the unique solution to the exponential equation dXt = Xt dMt , X0 = 1 If γ ∈ L(M ), then all solutions X to the equation dXt = γt Xt dMt ... available from the publisher.) Continuous Stochastic Calculus with Applications to Finance MICHAEL MEYER, Ph.D CHAPMAN & HALL/CRC Boca Raton London New York Washington, D.C Library of Congress.. .Continuous Stochastic Calculus with Applications to Finance APPLIED MATHEMATICS Editor: R.J Knops This series presents texts and monographs... J.) Continuous stochastic calculus with applications to finance / Michael Meyer p cm. (Applied mathematics ; 17) Includes bibliographical references and index ISBN 1-58488-234-4 (alk paper) Finance